A solid has as its base a circular region in the xy plane bounded by the graph of x^2 + y^2 = 4. Find the volume of a solid if every cross section by a plane perpendicular to the x-axis is an isosceles triangle with base on the xy plane and altitude equal to the length of the base.
To find the volume of the solid, we need to determine the area of each cross section and integrate it over the given range.
First, let's consider a cross section of the solid by a plane perpendicular to the x-axis. This cross section is an isosceles triangle with base on the xy plane and altitude equal to the length of the base.
Since the base of the isosceles triangle lies on the xy plane, it is a circle with a radius of 2 (given by the equation x^2 + y^2 = 4). The length of the base of the isosceles triangle is then equal to the diameter of this circle, which is 2 times the radius, or 4.
Now, let's determine the area of this isosceles triangle. The area of a triangle is given by the formula: A = (1/2) * base * height.
In this case, the base and height of the triangle are equal since it is an isosceles triangle with an altitude equal to the length of the base. So, the area of each cross section is given by: A = (1/2) * 4 * 4 = 8.
To find the volume of the solid, we need to integrate the area of each cross section over the given range. The range is determined by the x-values where the circle x^2 + y^2 = 4 intersects with the x-axis.
To find these x-values, we need to solve the equation x^2 + y^2 = 4 for y = 0 (since we are looking for points on the x-axis). This simplifies to x^2 + 0^2 = 4, which gives us x = ±2.
Therefore, the range of integration in the x-axis is from -2 to 2.
Finally, we can calculate the volume of the solid by integrating the area of each cross section over this range:
V = ∫[from -2 to 2] (8) dx
Integrating 8 with respect to x gives us:
V = [8x] from -2 to 2
V = 8(2) - 8(-2)
V = 16 + 16
V = 32
Therefore, the volume of the solid is 32 cubic units.